of 1 26
Abstract
Here I show that the
biological life elements
and artificial life elements
and their compounds can
have their behaviors
predicted as
mathematical constructs
not just as chemical
constructs and that they
describe one another.
Since the laws of Nature
are those things which
are invariant under
transformation and this
work is a mathematical
formulation not chemical,
then this mathematical
formulation in this paper
for the biological and AI
elements and their
compounds may find its
uses in understanding the
Natural laws governing
matter.
MATHEMATICAL CONSTRUCTS BETWEEN AI AND BIOLOGICAL LIFE
of 2 26
KEY TALKING POINTS
Prepared by IAN BEARDSLEY JUNE 28 2021
Table of Contents
Important……………………………3
The Computation…………………..4
The Dynamic Function…………….4
Gradient……………………………13
Divergence…………………………18
Appendix 1…………………………21
Appendix 2…………………………22
Appendix 3…………………………24
GENESIS PROJECT
of 3 26
Important
Above we see the artificial intelligence (AI) elements pulled out of the periodic table of the elements. As you
see we can make a 3 by 3 matrix of them and an AI periodic table. Silicon and germanium are in group 14
meaning they have 4 valence electrons and want 4 for more to attain noble gas electron configuration. If we
dope Si with B from group 13 it gets three of the four electrons and thus has a deficiency becoming positive
type silicon and thus conducts. If we dope the Si with P from group 15 it has an extra electron and thus
conducts as well. If we join the two types of silicon we have a semiconductor for making diodes and transistors
from which we can make logic circuits for AI.
As you can see doping agents As and Ga are on either side of Ge, and doping agent P is to the right of Si but
doping agent B is not directly to the left, aluminum Al is. This becomes important. I call (As-Ga) the
differential across Ge, and (P-Al) the differential across Si and call Al a dummy in the differential because
boron B is actually used to make positive type silicon.
That the AI elements make a three by three matrix they can be organized with the letter E with subscripts that
tell what element it is and it properties, I have done this:
Thus E24 is in the second row and has 4 valence electrons making it silicon (Si), E14 is in the first row and has
4 valence electrons making it carbon (C). I believe that the AI elements can be organized in a 3 by 3 matrix
makes them pivotal to structure in the Universe because we live in three dimensional space so the mechanics of
the realm we experience are described by such a matrix, for example the cross product. Hence this paper where
I show AI and biological life are mathematical constructs and described in terms of one another.
We see, if we include the two biological elements in the matrix (E14) and and (E15) which are carbon and
nitrogen respectively, there is every reason to proceed with this paper if the idea is to show not only are the AI
elements and biological elements mathematical constructs, they are described in terms of one another. We see
this because the first row is ( B, C, N) and these happen to be the only elements that are not core AI elements in
E
13
E
14
E
15
E
23
E
24
E
25
E
33
E
34
E
35
GENESIS PROJECT
of 4 26
the matrix, except boron (B) which is out of place, and aluminum (Al) as we will see if a dummy
representative, makes for a mathematical construct, the harmonic mean. Which means we have proved our case
because the first row if we take the cross product between the second and third rows are, its respective unit
vectors for the components, meaning they describe them!
The Computation
And silicon (Si) is at the center of our AI periodic table of the elements. We see the biological elements C and
N being the unit vectors are multiplied by the AI elements, meaning they describe them! But we have to ask;
Why does the first row have boron in it which is not a core biological element, but is a core AI element? The
answer is that boron is the one AI element that is out of place, that is, aluminum is in its place. But we see this
has a dynamic function.
The Dynamic Function
The primary elements of artificial intelligence (AI) used to make diodes and transistors, silicon (Si) and
germanium (Ge) doped with boron (B) and phosphorus (P) or gallium (Ga) and arsenic (As) have an
A = (Al, Si, P )
B = (G a, G e, As)
A ×
B = − 145
B + 138
C + 1.3924
N
A = 26.98
2
+ 28.09
2
+ 30.97
2
= 50g /m ol
B = 69.72
2
+ 72.64
2
+ 74.92
2
= 126g /m ol
A ⋅
B = A Bcosθ
cosθ =
6241
6300
= 0.99
θ = 8
∘
A ×
B = A Bsi nθ = (50)(126)sin8
∘
= 877.79
877.79 = 29.6g /m ol ≈ Si = 28.09g /m ol
of 5 26
asymmetry due to boron. Silicon and germanium are in group 14 like carbon (C) and as such have 4 valence
electrons. Thus to have positive type silicon and germanium, they need doping agents from group 13 (three
valence electrons) like boron and gallium, and to have negative type silicon and germanium they need doping
agents from group 15 like phosphorus and arsenic. But where gallium and arsenic are in the same period as
germanium, boron is in a different period than silicon (period 2) while phosphorus is not (period 3). Thus
aluminum (Al) is in boron’s place. This results in an interesting equation.
The differential across germanium crossed with silicon plus the differential across silicon crossed with
germanium normalized by the product between silicon and germanium is equal to the boron divided by the
average between the germanium and the silicon. The equation has nearly 100% accuracy (note: using an older
value for Ge here, is now 72.64 but that make the equation have a higher accuracy):
Si(A s − G a) + G e(P − Al )
SiG e
=
2B
Ge + Si
28.09(74.92 − 69.72) + 72.61(30.97 − 26.98)
(28.09)(72.61)
=
2(10.81)
(72.61 + 28.09)
0.213658912 = 0.21469712
0.213658912
0.21469712
= 0.995
of 6 26
Due to an asymmetry in the periodic table of the elements due to boron we have the harmonic mean
between the semiconductor elements (by molar mass):
This is Stokes Theorem if we approximate the harmonic mean with the arithmetic mean:
We can make this into two integrals:
If in the equation (The accurate harmonic mean form):
We make the approximation
Then the Stokes form of the equation becomes (See Appendix 2)
Which can be generalized to include the AI elements by density and atomic radius:
Si
B
(As − G a) +
Ge
B
(P − Al ) =
2SiG e
Si + G e
∫
S
( ∇ × u ) ⋅ d S =
∮
C
u ⋅ d r
∫
1
0
∫
1
0
[
Si
B
(As − G a) +
Ge
B
(P − Al )
]
d xd y ≈
1
Ge − Si
∫
Ge
Si
x d x
∫
1
0
∫
1
0
Si
B
(As − G a)d yd z ≈
1
3
1
(Ge − Si )
∫
Ge
Si
x d x
∫
1
0
∫
1
0
Ge
B
(P − Al ) d x dz ≈
2
3
1
(Ge − Si )
∫
Ge
Si
yd y
Si
B
(As − G a) +
Ge
B
(P − Al ) =
Ge − Si
Ge
Si
dx
x
2SiGe
Si + Ge
≈ Ge − Si
∫
1
0
∫
1
0
[
Si
B
(As − G a) +
Ge
B
(P − Al )
]
d yd z =
∫
Ge
Si
d x
of 7 26
(See Appendix 1). Thus we see for this approximation there are two integrals as well:
For which the respective paths are
By making the approximation
In
We have (See Appendix 3)
Which is Ampere’s Circuit Law
∇ ⋅ Q = C f
−1
(
1
n
⋅
n
∑
i=1
f (x
i
)
)
∫
1
0
∫
1
0
Si
B
(As − G a)d yd z =
1
3
∫
Ge
Si
dz
∫
1
0
∫
1
0
Ge
B
(P − Al ) d ydz =
2
3
∫
Ge
Si
dz
y
1
=
1
3
B
SiGa
ln(z)
y
2
=
2
3
B
Si Al
ln(z)
2SiGe
Si + Ge
≈ Ge − Si
Si(As − G a)
B
+
Ge(P − Al )
B
=
2SiGe
Si + Ge
Si
ΔGe
ΔS
+ G e
ΔSi
ΔS
= B
of 8 26
We see if written
Which is interesting because it is semiconductor elements by molar mass, which are used to make
circuits. This results in an interesting constant C which is the golden ratio conjugate phi divided by
the semiconductor elements if we take the square root of it (See Appendix 2):
Where (Phi) is given by
and
And
=1.618
=0.618
(phi) the golden ratio conjugate. We also find
Thus since
And we have
∇ ×
B = μ
0
J + μ
0
ϵ
∂
E
∂t
Si
ΔGe
ΔS
= B − Ge
ΔSi
ΔS
C =
ϕ
Si + Ge
Φ
a = b + c
a
b
=
b
c
Φ = a /b
ϕ = b /a
ϕ
(ϕ)ΔGe + (Φ)ΔSi = B
∇ ×
B = μ
0
J + μ
0
ϵ
∂
E
∂t
Si
ΔGe
ΔS
= B − Ge
ΔSi
ΔS
of 9 26
We see and are both and c is in the Si (silicon) field wave, but for E and B fields c is the speed of light.
ΔGe =
ΔS
Si
B −
Ge
Si
ΔSi
(
∇
2
−
1
c
2
∂
2
∂t
)
E = 0
(
∇
2
−
1
c
2
∂
2
∂t
)
B = 0
c =
1
ϵu
0
≈ ϕ
μ
ϵ
0
Φ
ϕ
ϵ = 8.854E − 12F ⋅ m
−1
μ
0
= 1.256E − 6H /m
Ge
Si
= μ
0
ϵ
ΔS
Si
= μ
0
Si
Ge ⋅ ΔS
=
28.09
(72.64)(44.5)
=
1
2
of 10 26
Which is a sawtooth (See Appendix 3)
(
∇
2
−
1
c
2
∂
2
∂x
)
Si = 0
(
∇
2
−
1
c
2
∂
2
∂x
)
Ge = 0
of 11 26
of 12 26
This is the halfwave
Interestingly, the 0.4 is boron (B) over aluminum (Al) the very two elements that lead us to looking for a wave
equation because boron was the out of place element in the AI periodic table that lead to us using aluminum as
its dummy representative in the Si differential and that itself divided into the left hand terms to give us the
harmonic mean between the central AI elements semiconductor materials Si and Ge. The Ag and Cu are the
central malleable, ductile, and conductive metals used in making electrical wires to carry a current in AI
circuitry.
y(x) = e
0.4x
+ 1.7
y(x) = e
2
5
x
+
17
10
y(x) = e
B
Al
x
+
Ag
Cu
B
Al
=
10.81
26.98
= 0.400667
Ag
Cu
=
107.87
63.55
= 1.6974 ≈ 1.7
of 13 26
Gradient
We have the vector field!
!
And we want to find the scalar field such that . We see is a conservative vector
field because!
!
Thus to find it!
!
!
!
!
Earlier we had!
u =
(
0,
Si
B
(As)y,
Si
B
(Ga)z
)
ϕ
u = ∇ϕ
u ( r )
i
j
k
∂
∂x
∂
∂y
∂
∂z
0
Si
B
(As)y
Si
B
(Ga)z
= 0
∂ϕ
∂x
= 0
∂y
∂ϕ
=
Si
B
(As)y
∂z
∂ϕ
=
Si
B
(Ga)z
ϕ =
Si
2B
(As)y
2
+
Si
B
(Ga)z
2
u =
(
0,
Si
B
(Ga)z,
Si
B
(As)y
)
(
0,
Si
B
(Ga)z,
Si
B
(As)y
)
⋅ (d x, d y, dz) =
1
3
x d x
Ge − Si
d r = (d x, d y, d z)
of 14 26
Let us change x to z on the right:
Then,..
,
Is a line in the y-z plane (See Appendix 2).
Thus we have…
(
0,
Si
B
(Ga)z,
Si
B
(As)y
)
⋅ (d x, d y, dz) =
1
3
zdz
Ge − Si
d x = 0
dz = 0
d y =
Bd z
3(Si )(Ga)(Ge − Si )
r = (t)
j + (26,178t)
k
y =
10.81
3(28.09)(69.72)(72.64 − 28.09)
z
y = Cz
C = 0.0000382
mol
2
g
2
of 15 26
of 16 26
Thus applying the gradient theorem to our AI elements we look at the analog to the electric potential
in electrodynamics. Since Coulombs law is:
Then,..
We find that
I calculate that doping silicon with boron, or phosphorus, etc (next page) that there are about
19,241.72 grams Si per gram of boron. Thus we have:
Which is on the order of the masses of carbon (12.01) and nitrogen (14.01) which is where it should
be if our equation
Is the integral over a path of some kind of potential that determines the amount of doping agent per
semiconductor material.
F = k
e
q
1
q
s
r
2
k
e
= 8.988E 9N ⋅ m ⋅ C
−2
=
1
4πϵ
0
V
E
=
1
4πϵ
0
Q
r
ϵ
0
= 8.54E − 12F ⋅ m
−1
∫
Ge−Si
Si
∇ϕ ⋅ d r = 2,835,470602.78
g
3
mol
3
3
2,835,470602.78 = 1415.3865g /m ol
19,241.72
1,415.3865
= 13.59
g
mol
∫
Ge−Si
Si
∇ϕ ⋅ d r =
(
Si
B
(As)
t
2
2
+ 26,178
Si
B
(Ga)
t
2
2
)
{
Ge − Si
Si
of 17 26
of 18 26
Divergence
So far everything has lined up with mathematical theorems and their counterparts in electrodynamics to suggest
that biological life and artificial intelligence are mathematical constructs that describe one another. Owing to
the unusual placement of boron in what we I have called the AI Periodic Table of the Elements, and using
aluminum in its place we have shown that Stoke’s theorem follows, and that Ampere’s Circuit law with
Maxwell’s addition, which is one of Maxwell’s equations, follows and also that a wave equation follows if we
are to parallel the laws of electricity and Magnetism and that if we take for it the exponential e, we raise it to B/
Al, because B and Al are the components of the theory that give rise to its dynamics and that this works very
accurately. We have proceeded to derive a potential to continue with idea and we have seen it predicts the
amount of doping agents use to dope the semiconductor element silicon. It only remains to find the counterpart
for the divergence theorem which relates a surface area to its volume and is Gauss’s law (one of the Maxwell
Equations) which is:
The flux of an electric field through any closed surface is the net charge Q over . For a point charge enclosed
in a sphere we get Coulombs law, and an inverse square field. In other words, from a point charge the field
diverges.
We arrived at our wave
By saying (not literally, not equals, but analogous to)
Where
Very accurately. We have that
∮
E ⋅ d
a =
Q
ϵ
0
ϵ
0
y(x) = e
B
Al
x
+
Ag
Cu
G e
Si
= μϵ
0
c =
1
ϵ
0
μ
B
Al
=
10.81
26.98
=
2
5
of 19 26
And,…
We can say
Thus,..
Thus,…
Thus,…
V
E
=
1
4π ϵ
0
Q
r
ϕ =
Si
2B
(As)y
2
+
Si
B
(G a)z
2
Δ S
Si
=
G e − Si
Si
= μ
μ =
72.64 − 28.09
28.09
= 1.58697
ϵ
0
=
G e
Si
1
μ
=
72.64
28.09
1
1.587
= 1.63 ≈ Φ
1
4π (1.63)
Q
r
2
= F
4π (1.63) = 20.5
0.05 =
r
2
F
Q
0.05Q = F ⋅ r
2
∮
d
a = 0.05
Q
F
∮
F ⋅ d
a = 0.05Q
of 20 26
Conclusion
Thus we see that the behavior of biological elements and their compounds can be manipulated and predicted as
mathematical constructs, not just by the laws of chemistry. And that the biological and artificial life describe
one another.
of 21 26
Appendix 1
The power mean is obtained by letting
It is the geometric mean if
and by molar mass or p=-1 (harmonic mean)
and by density p=0 (geometric mean) or
and by atomic radius p=1 (arithmetic mean) .
Are respectively and
And, the ratios
and by molar mass or
and by density or
and by atomic radius where C is
Are, quotients , and , respectively, then
∇ ⋅ Q = C f
−1
(
1
n
⋅
n
∑
i=1
f (x
i
)
)
f (x) = x
p
f (x) = log(x)
(As − G a)
(P − Al )
(Al − P)
(Al − P)
(Al − P)
(Ga − As)
C = Φ
ΔE
1
ΔE
2
Si
B
Ge
B
B
2Ge(Ga − As)
Si
B
2Ge(Ga − As)
P
Si
B
Ge
B
Φ
Q
1
Q
2
∇ = (ΔE
1
, ΔE
2
)
Q = (Q
1
, Q
2
)
of 22 26
Appendix 2
The geometric interpretation is as follows…
of 23 26
Let us change x to z on the right:
,
Is a line in the y-z plane.
0.618 is exactly the the golden ratio conjugate. Thus the square root of C is precisely this value
reduced by a factor of 100. That is…
At this point it would be good to note that Si+Ge=100.73 g/mol, almost exactly 100.
u =
(
0,
Si
B
(Ga)z,
Si
B
(As)y
)
(
0,
Si
B
(Ga)z,
Si
B
(As)y
)
⋅ (d x, d y, dz) =
1
3
x d x
Ge − Si
d r = (d x, d y, d z)
(
0,
Si
B
(Ga)z,
Si
B
(As)y
)
⋅ (d x, d y, dz) =
1
3
zdz
Ge − Si
d x = 0
dz = 0
d y =
Bd z
3(Si )(Ga)(Ge − Si )
r = (t)
j + (26,178t)
k
y =
10.81
3(28.09)(69.72)(72.64 − 28.09)
z
y = Cz
C = 0.0000382
mol
2
g
2
C = 0.00618m ol /g
C =
ϕ
100
of 24 26
Appendix 3
of 25 26
is the differential across Si, is the differential across Ge
is the vertical differential.!
The equation of a sawtooth is!
!
Which rises as a straight line and the drops straight down and then starts over again. The straight
line that rises can reduce this equation to!
!
I find we can approximate our curve the differentials across period 14 with a sawtooth given by!
!
Which can be approximated with phi ( ) the golden ratio conjugate:!
!
Our differentials are!
It is amazing how accurately we can fit these differentials with and exponential equation for the upward
increase. The equation is
ΔSi = P − Al
ΔGe = As − G a
ΔS = Ge − Si
y(x) = −
2a
π
tan
−1
(
cot
xπ
p
)
y(x) =
2a
p
y(x) =
8
5
x + 1
ϕ
y(x) = ϕx + 1
ΔC = N − B = 14.01 − 10.81 = 3.2
Δ Si = P − Al = 30.97 − 26.98 = 3.99
ΔG e = A s − G a = 74.92 − 69.72 = 5.2
Δ Sn = Bi − In = 121.75 − 114.82 = 6.93
ΔPb = Bi − T l = 208.98 − 204.38 = 4.6
y(x) = e
0.4x
+ 1.7
y(x) = e
2
5
x
+
17
10
of 26 26
Ian Beardsley